🌡️ Temperature Distribution

Visualise how temperature distributes along rods, fins, and slabs under various boundary conditions. Compare numerical (FDM) solutions with exact analytical answers, and observe transient temperature evolution over time.

🇺🇦 Українська

Mode

Boundary & Properties

Statistics

T mean—
T max—
T min—
Fourier # Fo—
Numerical
Analytical

Heat Transfer Theory

Steady-state (no heat source): T(x) = TL + (TR−TL)·x/L — linear profile. With uniform volumetric heat source Q: T(x) = TL + (TR−TL)·x/L + Q/(2k)·x(L−x) — parabolic. Transient: ∂T/∂t = α·∂²T/∂x² solved by explicit FDM. The Fourier number Fo = α·t/L² measures dimensionless time; at Fo ≈ 0.2 the solution approaches steady-state. Extended surface (fin): d²θ/dx² − m²θ = 0, m² = hP/(kA), θ = T−T∞, giving an exponential solution θ(x) = θb·cosh[m(L−x)]/cosh(mL).

About this simulation

This simulation solves the 1D heat conduction equation along a rod under three regimes: Steady (equilibrium temperature profile), Transient (time-evolving diffusion from an explicit finite-difference solver), and Fin (a rod losing heat to its surroundings via convection). Every numerical curve is drawn alongside its closed-form analytical solution so you can see exactly how well a discretised model tracks the exact physics.

🔬 What it shows

Steady mode plots the linear analytical profile T(x) between T_left and T_right with an optional heat source Q. Transient mode steps the explicit FDM scheme ∂T/∂t = α·∂²T/∂x² forward in time and overlays a Fourier-series analytical solution. Fin mode uses the exponential cooling-fin solution governed by the Biot number Bi.

🎮 How to use

Switch modes with the Steady / Transient / Fin buttons. Drag T_left and T_right to set boundary temperatures, Source Q to add internal heating or cooling, Diffusivity α to change how fast heat spreads, and Biot Bi (fin mode) to set convective loss strength. Reset restarts the run and the live Fourier number Fo tracks dimensionless elapsed time.

💡 Did you know?

The Fourier number Fo = α·t/L² is the key dimensionless group in transient conduction: once Fo exceeds roughly 0.2-0.3, a rod is close enough to its steady-state profile that higher Fourier-series terms become negligible — this is why engineers use Fo to judge whether a "quasi-steady" approximation is safe.

Frequently asked questions

What is the difference between the numerical and analytical curves?

The numerical curve comes from an explicit finite-difference method (FDM) stepping the heat equation forward in small time increments on a discretised grid. The analytical curve is the exact closed-form solution (linear for steady state, a Fourier series for transient, exponential for a fin). Comparing them shows how closely a numerical scheme approximates the true physics.

What does the Fourier number Fo represent?

Fo = α·t/L² is a dimensionless measure of elapsed time relative to the diffusion timescale L²/α. Small Fo means heat has barely diffused from its initial distribution; Fo approaching 1 or beyond means the rod has essentially reached its steady-state temperature profile.

Why does the Biot number matter in Fin mode?

The Biot number Bi compares convective heat loss at the fin's surface to conductive heat flow inside it. Low Bi means conduction dominates and the fin stays nearly uniform in temperature; high Bi means the fin loses heat to its surroundings quickly and its temperature drops sharply along its length.

What happens if I set Source Q away from zero?

A nonzero Q adds uniform internal heat generation (positive) or removal (negative) along the rod, which bends the steady-state temperature profile from a straight line into a parabola — the same effect seen in current-carrying wires (Joule heating) or nuclear fuel rods.

Why does increasing diffusivity α speed up the transient response?

α = k/(ρc) measures how quickly heat spreads through a material relative to how much energy it takes to change its temperature. A higher α means the same explicit FDM time step produces faster equilibration, so the transient curve converges to the steady-state profile in fewer simulated seconds.